Note on Completely Monotone Densities

In [2] it is proved that mixtures of exponential distributions are infinitely divisible (id). In [3] it is proved that the same holds for the discrete analogue, i.e. for mixtures of geometric distributions. In this note we show that these results imply that a density function $f(x)$ (or distribution ₙ\ on the integers) is id if the function $f(x)$ (or the sequence ₙ\ is completely monotone (cm). For the definition and properties of cm functions and sequences we refer to [1].